Lest my statement be interpreted too strongly, let me say that I have not personally verified the contents. They may contain errors. However, the thoroughness and clarity of the exposition lead me to the confidence that I can replicate and verify the results, as can anyone else. When I am ready, I will check their work because they wrote down enough for me to do so. Gerhard "Then I'll Build The Bridge" Paseman, 2015.07.26

Wow @Gerry, thanks for the link! I will peruse the paper more closely, but on first skimming, I don't see any remarks that say the paper's results apply to $\sigma$. I do find a 1991 paper on $\omega$ that looks interesting though. Gerhard "Deeper Into The Rabbit Hole" Paseman, 2015.07.21

@zeraouliarafik , I recommend that you no longer comment on this answer; it is getting too long to follow. Also, you should check that such n do not satisfy your condition: \sigma(114^(114k)) is odd. Further, I do not want to be emailed on conjectural statements. You should show that you put in some work to resolve a question you have. Gerhard "Look At Twice A Square" Paseman, 2015.07.20

Using a regular n-gon for reference, I find the diagonals closest to the center the most restrictive, suggesting that m will be closer to (perhaps greater than) n than to n/2 for large n. Gerhard "But Geometry Might Help Though" Paseman, 2015.07.19

I deleted my comment because I thought diagonals were also not to be shared. However, I thought a sub-polygon of a convex polygon was also convex (I guess vertex order matters?). There may still be a design interpretation, but my original one does not quite capture the problem. Gerhard "Thinks This Still Isn't Geometry" Paseman, 2015.07.19

It might clarify things to note that this is "happening inside S", where S is the set of squarefree numbers and has a natural semilattice structure on it. Z is just X intersect the complement of a join-ideal of S generated by those members of Y that are not primes. If there are finitely many nonprime members of Y, it may be straightforward to show the ideal grows slowly. It may also be possible to find a "sparse infinite cover" which would force Z to be much smaller than you hope. Gerhard "This Mindset Worked Once Before" Paseman, 2015.07.17

I recommend further discussion in email, not comments. Also, it is unclear how the paper relates to the problem. Iterates of the totient function are not necessarily related to iterates of the sigma function. Gerhard "You Can Explain In Email" Paseman, 2015.07.16

I think you should not accept this version. I am thinking about a proof, as well as related questions (which you should think about as well). If there is no further progress, I will add a further edit which says what difficulties there are in showing that such a sequence exists. If you find the result acceptable, it would be OK with me to accept that future version. Gerhard "Now Back To The Present" Paseman, 2015.07.16

Ah yes. I was not paying enough attention. Ideal would be appropriate, although I might still prefer "absorbing set". However, the literature I know talks only about things like left-absorbing elements b (ab=b). Gerhard "Seeks Something Even More General" Paseman, 2015.07.16

I prefer "closed under convolution". If the set of functions combined with convolution were an algebra (commutative semigroup?), you would be looking for the subalgebras of this universal algebra. Gerhard "Do You Seek This Generality?" Paseman, 2015.07.16

@zeraouliarafik: please delete your last comment. I present my personal information the way I do to keep a low profile. Gerhard "Doesn't Need Spambots Knowing It" Paseman, 2015.07.15

@zeraouliarafik for permission to email me on this, you have it. For the email address, you have to do (a small amount of) research. Gerhard "Hint: Try The User Page" Paseman, 2015.07.15

@zeraouliarafik, I'm not sure. I would be happy to be acknowledged for my contributions should you wish to write such a paper. I can be pretty picky about style in something that has my name as author though. However, people have studied iterates of $\sigma()$ before; If you want my help with such a paper, my first question would be to get the bibliography of papers you have read on the subject, and my second would be for the list of papers you have skimmed or intend to skim. Gerhard "Acknowledging The Literature Is Important" Paseman, 2015.07.15

@zeraouliarafik But that is a different question: you are now asking for an integer m and integer x such that $x$ divides $\sigma^k(m)$ for all $k$. It may be possible that 2 is such an integer, but even that is not known. Gerhard "This Question Is Hard Enough" Paseman, 2015.07.15

It seems to me that one should be able to construct an injection into "halves" of $\omega$ (infinite sets with infinite complements) from the set of (n-1) subsets of $\omega$ so that each finite subset is a subset of its image and conditions 2) and 3) are held. Gerhard "Is This Ramsey In Disguise?" Paseman, 2015.07.15